Monocular Depth Estimation via Neural Network with Learnable Algebraic Group and Ring Structures

TL;DR

LAGRNet introduces an algebraic geometry grounded framework for monocular depth estimation, embedding learnable group, ring, and sheaf structures to improve accuracy and generalization.

Contribution

It is the first to incorporate algebraic geometric structures into deep learning for monocular depth estimation, enhancing robustness and cross-scale consistency.

Findings

Outperforms state-of-the-art methods on KITTI, NYU-Depth V2, ETH3D benchmarks.

Achieves significant accuracy improvements in zero-shot evaluations.

Demonstrates robustness to view changes and cross-scale variations.

Abstract

Monocular depth estimation (MDE) has witnessed remarkable progress driven by Convolutional Neural Networks and transformer-based architectures. However, these approaches typically treat the problem as a generic image-to-image regression on Euclidean grids, thereby overlooking the intrinsic algebraic and geometric structures induced by perspective projection. To address this limitation, we propose LAGRNet, a novel framework that fundamentally grounds MDE in algebraic geometry by explicitly embedding learnable group, ring, and sheaf structures into the deep learning pipeline. Modeling feature maps as sections of a sheaf over an approximated image manifold, our method first establishes a Group-defined Feature Manifold (GFM) parameterized by a learned algebraic group action to enforce projective equivariance and robustness against view changes. To facilitate algebraically consistent cross-scale interactions, we subsequently introduce a Ring Convolution Layer (RCL) that formulates feature fusion as a graded ring homomorphism. Furthermore, to ensure global topological consistency, a Sheaf-based Module (SM) aggregates local depth cues via \v{C}ech nerve on the image topology. Extensive zero-shot evaluations across the KITTI, NYU-Depth V2, and ETH3D benchmarks demonstrate that LAGRNet significantly outperforms state-of-the-art methods in both accuracy and generalization capabilities.